Additive problems on $\lfloor p^c\rfloor$

The sequence $${\mathbb{P}}^{(c)}=(\lfloor p^c\rfloor {)}_{p\in \mathbb{P}}(c>0,c\mathrm{\notin }\mathbb{N}),$$ is an important subsequence of the well-known Piatetski-Shapiro sequence, where $\mathbb{P}$ is the set of prime numbers and $\lfloor \cdot \rfloor$ is the floor function. We prove that for all $c\in (0,13\mathrm{/}15)$, any large enough integer $N$ can be represented as $$N=\lfloor p^c\rfloor +q,$$ where $p$ and $q$ are primes. We also prove the result holds for almost all fixed positive $c\in \mathbb{R}\setminus \mathbb{Z}$. Moreover, we investigate shifted primes in this sequence, obtaining an asymptotic formula for all $c\in (0,13\mathrm{/}15)$ and an almost-all result for fixed positive $c\in \mathbb{R}\setminus \mathbb{Z}$.